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<span class="kw">let </span><font color="Maroon" title="c1">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font><br/>  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b4">G1</font> being    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">w</font><br/>  for <font color="Olive" title="b5">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b4">G1</font>  st  not <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b5">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</font> holds <br/>( <font color="Olive" title="b5">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b5">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">w</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b5">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Olive" title="b5">w1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Olive" title="b5">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b5">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">w</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">w</font><br/>  for <font color="Olive" title="b3">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">G1</font>  st  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">w</font> holds <br/>( <font color="Olive" title="b3">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b3">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b3">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Olive" title="b3">w1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Olive" title="b3">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b3">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G1</font> being    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">w</font><br/>  for <font color="Olive" title="b2">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G1</font>  st  not <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">w</font> holds <br/>( <font color="Olive" title="b2">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b2">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Olive" title="b2">w1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Olive" title="b2">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">G1</font> be    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">w</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">G1</font>  st  not <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b1">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">w</font> holds <br/>( <font color="Olive" title="b1">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b1">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Olive" title="b1">w1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Olive" title="b1">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">w1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">G1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">w</font> implies ( <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
(  not <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">w</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/>

<a NAME="E2:54"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c1">G2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T57" onmouseover="rs('glib_006/T57')" onmouseout="rh()">GLIB_006:57</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  implies <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  implies <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; ( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E1:54_1"/>
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  implies <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:54_1_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T57" onmouseover="rs('glib_000/T57')" onmouseout="rh()">GLIB_000:57</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_1_1_1"/>
( <font color="Maroon" title="c7">f</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c1">G2</font> or  not <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="glib_006.html#T71" onmouseover="rs('glib_006/T71')" onmouseout="rh()">GLIB_006:71</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:54_1_1_1_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c1">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:54_1_1_1_1"/>
<font color="Maroon" title="c7">f</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><a NAME="E3:54_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E1:54_1_1_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="glib_000.html#T57" onmouseover="rs('glib_000/T57')" onmouseout="rh()">GLIB_000:57</a></span>;<br/><span class="kw">hence </span><a NAME="E4:54_1_1_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:54_1_1_1_2"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
 not <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:54_1_1_1_2"/>
<font color="Maroon" title="c7">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E3:54_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E1:54_1_1_1_2"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="glib_006.html#T106" onmouseover="rs('glib_006/T106')" onmouseout="rh()">GLIB_006:106</a></span>;<br/><a NAME="E4:54_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:54_1_1_1_2"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:54_1"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  or <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:54_1_2_1"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:54_1_2_1"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="glib_000.html#T78" onmouseover="rs('glib_000/T78')" onmouseout="rh()">GLIB_000:78</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:54_1_2_2"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span><br/><div class="add"><a NAME="E2:54_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:54_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="glib_006.html#T105" onmouseover="rs('glib_006/T105')" onmouseout="rh()">GLIB_006:105</a></span>;<br/><a NAME="E4:54_1_2_2"/>
<font color="Maroon" title="c7">f</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:54_1_2_2"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:54_1_2_2"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="glib_000.html#T57" onmouseover="rs('glib_000/T57')" onmouseout="rh()">GLIB_000:57</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:54"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/>

<a NAME="E5:54"/>
 not <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/>
<a NAME="E6:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T50" onmouseover="rs('zfmisc_1/T50')" onmouseout="rh()">ZFMISC_1:50</a></span>;<br/>
<span class="kw">thus </span><span class="lab"><font color="Green" title="E11">A9</font></span>: <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  = 
<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="card_2.html#T35" onmouseover="rs('card_2/T35')" onmouseout="rh()">CARD_2:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/>

<a NAME="E8:54"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="glib_000.html#T78" onmouseover="rs('glib_000/T78')" onmouseout="rh()">GLIB_000:78</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  implies <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:54_3"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:54_3"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T59" onmouseover="rs('glib_000/T59')" onmouseout="rh()">GLIB_000:59</a></span>;<br/><a NAME="E4:54_3"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_3_1"/><span class="lab"><font color="Green" title="E14">A13</font></span>: 
 not <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:54_3_1"/>
<font color="Maroon" title="c7">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_3"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<a NAME="E3:54_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E1:54_3_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="glib_006.html#T106" onmouseover="rs('glib_006/T106')" onmouseout="rh()">GLIB_006:106</a></span>;<br/>
<a NAME="E4:54_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="glib_006.html#T105" onmouseover="rs('glib_006/T105')" onmouseout="rh()">GLIB_006:105</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:54_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:54_3"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="glib_000.html#T125" onmouseover="rs('glib_000/T125')" onmouseout="rh()">GLIB_000:125</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:54_3"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:54_3"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="glib_000.html#T59" onmouseover="rs('glib_000/T59')" onmouseout="rh()">GLIB_000:59</a>, <a class="ref" href="glib_006.html#T71" onmouseover="rs('glib_006/T71')" onmouseout="rh()">GLIB_006:71</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:54"/><span class="kw">then </span>
<font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:54"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
<font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/>

<span class="kw">hence </span><a NAME="E12:54"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/>

<span class="kw">thus </span><a NAME="E13:54"/>
<font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="ref" href="xboole_1.html#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1</span><br/>

<span class="kw">thus </span><a NAME="E14:54"/>
<font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="txt" href="#E12:54"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="card_2.html#T19" onmouseover="rs('card_2/T19')" onmouseout="rh()">CARD_2:19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
